Iterative Algorithm: Difference between revisions

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An [[Iterative Algorithm]] is an [[algorithm]] that iterates between two or more [[algorithm step]]s.
An [[Iterative Algorithm]] is an [[algorithm strategy]] that iterates between two or more [[algorithm step]]s.
* <B><U>AKA</U>:</B> [[Iterative]].
* <B>Context</U>:</B>
* <B><U>Context</U>:</B>
** It can be applied by an [[Iterative Software Program]] (that can solve an [[iterative task]]).
** It can be associated with the solving of an [[Iterative Task]].
** It can range from being a [[Single-Batch Iterative Algorithm]] to being a [[Parallel Iterative Algorithm]].
* <B><U>Example(s)</U>:</B>  
* <B>Example(s):</B>  
** [[EM Algorithm]].
** [[EM Algorithm]].
** any [[Greedy Algorithm]].
** any [[Greedy Algorithm]].
* <B><U>Counter-Example(s)</U>:</B>  
** an [[Iterative Optimization Algorithm]].
** …
* <B>Counter-Example(s):</B>  
** A [[Recursive Algorithm]].
** A [[Recursive Algorithm]].
* <B><U>See</U>:</B> [[Algorithm Strategy]], [[Dynamic Programming Algorithm]], [[Iterative Game]].
* <B>See:</B> [[Sequential Algorithm]], [[Algorithm Strategy]], [[Dynamic Programming Algorithm]], [[Iterative Game]], [[Distributed Algorithm]].
 
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==References==


===2012===
== References ==
 
=== 2015 ===
* (Wikipedia, 2015) ⇒ http://wikipedia.org/wiki/Iteration#Computing Retrieved:2015-7-9.
** '''Iteration</B> in computing is the repetition of a block of statements within a [[computer program]]. It can be used both as a general term, synonymous with repetition, and to describe a specific form of repetition with a [[Mutable object|mutable]] state.  Confusingly, it may also refer to any repetition stated using an explicit repetition structure, regardless of mutability.
 
=== 2012 ===
* http://en.wikipedia.org/wiki/Iterative_method
* http://en.wikipedia.org/wiki/Iterative_method
** QUOTE: In [[computational mathematics]], an '''iterative method''' is a mathematical procedure that generates a sequence of improving approximate solutions for a class of problems. A specific implementation of an iterative method, including the [[Algorithm#Termination|termination]] criteria, is an [[algorithm]] of the iterative method. An iterative method is called '''convergent''' if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, [[heuristic]]-based iterative methods are also common.   <P>   In the problems of [[root-finding algorithm|finding the root]] of an equation (or a solution of a system of equations), an iterative method uses an initial guess to generate successive [[approximation]]s to a solution. In contrast, '''direct methods''' attempt to solve the problem by a finite  sequence of operations. In the absence of [[rounding error]]s, direct methods would deliver an exact solution (like solving a linear system of equations ''Ax'' = ''b'' by [[Gaussian elimination]]). Iterative methods are often the only choice for [[nonlinear equation]]s. However, iterative methods are often useful even for linear problems involving a large number of variables (sometimes of the order of millions), where direct methods would be prohibitively expensive (and in some cases impossible) even with the best available computing power.
** QUOTE: In [[computational mathematics]], an '''iterative method</B> is a mathematical procedure that generates a sequence of improving approximate solutions for a class of problems. A specific implementation of an iterative method, including the [[Algorithm#Termination|termination]] criteria, is an [[algorithm]] of the iterative method. An iterative method is called <B>convergent</B> if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, [[heuristic]]-based iterative methods are also common.       <P>           In the problems of [[root-finding algorithm|finding the root]] of an equation (or a solution of a system of equations), an iterative method uses an initial guess to generate successive [[approximation]]s to a solution. In contrast, '''direct methods</B> attempt to solve the problem by a finite  sequence of operations. In the absence of [[rounding error]]s, direct methods would deliver an exact solution (like solving a linear system of equations ''Ax'' = ''b'' by [[Gaussian elimination]]). Iterative methods are often the only choice for [[nonlinear equation]]s. However, iterative methods are often useful even for linear problems involving a large number of variables (sometimes of the order of millions), where direct methods would be prohibitively expensive (and in some cases impossible) even with the best available computing power.
 
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[[Category:Concept]]
[[Category:Concept]]

Latest revision as of 18:39, 17 September 2021

An Iterative Algorithm is an algorithm strategy that iterates between two or more algorithm steps.



References

2015

  • (Wikipedia, 2015) ⇒ http://wikipedia.org/wiki/Iteration#Computing Retrieved:2015-7-9.
    • Iteration in computing is the repetition of a block of statements within a computer program. It can be used both as a general term, synonymous with repetition, and to describe a specific form of repetition with a mutable state. Confusingly, it may also refer to any repetition stated using an explicit repetition structure, regardless of mutability.

2012

  • http://en.wikipedia.org/wiki/Iterative_method
    • QUOTE: In computational mathematics, an iterative method is a mathematical procedure that generates a sequence of improving approximate solutions for a class of problems. A specific implementation of an iterative method, including the termination criteria, is an algorithm of the iterative method. An iterative method is called convergent if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, heuristic-based iterative methods are also common.

      In the problems of finding the root of an equation (or a solution of a system of equations), an iterative method uses an initial guess to generate successive approximations to a solution. In contrast, direct methods attempt to solve the problem by a finite sequence of operations. In the absence of rounding errors, direct methods would deliver an exact solution (like solving a linear system of equations Ax = b by Gaussian elimination). Iterative methods are often the only choice for nonlinear equations. However, iterative methods are often useful even for linear problems involving a large number of variables (sometimes of the order of millions), where direct methods would be prohibitively expensive (and in some cases impossible) even with the best available computing power.